An Approach to Constructing “Good”Two-level Orthogonal Factorial Designs with

Large Run Sizesby

Chenlu Shi

B.Sc. (Hons.), St. Francis Xavier University, 2013

Project Submitted in Partial Fulfillment

of the Requirements for the Degree of

Master of Science

in the

Department of Statistics and Actuarial Science

Faculty of Science

c⃝ Chenlu Shi 2015

SIMON FRASER UNIVERSITY

Summer 2015

All rights reserved.However, in accordance with the Copyright Act of Canada, this work may be

reproduced without authorization under the conditions for “Fair Dealing.” Therefore,limited reproduction of this work for the purposes of private study, research, criticism,

review and news reporting is likely to be in accordance with the law, particularly ifcited appropriately.

Approval

Name: Chenlu Shi

Degree: Master of Science (Statistics)

Title: An Approach to Constructing “Good” Two-level Orthogo-nal Factorial Designs with Large Run Sizes

Examining Committee: Dr. Tim Swartz (chair)Professor

Dr. Boxin TangSenior SupervisorProfessor

Dr. Richard LockhartSupervisorProfessor

Dr. Liangliang WangInternal ExaminerAssistant Professor

Date Defended: 20 July 2015

ii

Abstract

Due to the increasing demand for two-level fractional factorials in areas of science and technology,it is highly desirable to have a simple and convenient method available for constructing optimalfactorials. Minimum G2-aberration is a popular criterion to use for selecting optimal designs.However, direct application of this criterion is challenging for large designs. In this project, wepropose an approach to constructing a “good” factorial with a large run size using two smallminimum G2-aberration designs. Theoretical results are derived that allow the word lengthpattern of the large design to be obtained from those of the two small designs. Regular 64-runfactorials are used to evaluate this approach. The designs from our approach are very close to thecorresponding minimum aberration designs, and they are even equivalent to the correspondingminimum aberration designs, when the number of factors is large.

Keywords: Fractional factorial; Minimum aberration; Minimum G2-aberration; Word lengthpattern.

iii

Dedication

To my family!

iv

Acknowledgements

I would like to express my deepest gratitude to my senior supervisor Dr. Boxin Tang for hisprofessional guidance, unlimited patience, and great mentoring throughout my graduate stud-ies. Without his encouragement and his insightful suggestions, this project would not have beenpossible. It has been such a privilege for me to be his student.

I would also like to thank the members of the examining committee, Dr. Richard Lockhartand Dr. Liangliang Wang, for their thorough reading of this project and valuable feedbacks. Iam grateful to all professors who taught me during the period of my graduate studies, includingDr. Jiguo Cao, Dr. Joan Hu, Dr. Brad McNeney and many others. I wish to thank Sadika,Kelly and Charlene for their kind help in the past two years. Many thanks to the Department ofStatistics and Actuarial Science at Simon Fraser University for offering such a fabulous graduateprogram.

Last but not least, I would like to thank my family, especially my parents for their uncondi-tional love and support.

v

Table of Contents

Approval ii

Abstract iii

Dedication iv

Acknowledgements v

Table of Contents vi

List of Tables vii

List of Figures viii

1 Introduction 11.1 Notation and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2 General Theoretical Results 52.1 Constructing A Large Design Using Two Small Designs . . . . . . . . . . . . . . 52.2 Doubling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 Small Design Without A Column of 1’s . . . . . . . . . . . . . . . . . . . . . . . 82.4 Small Design with A Column of 1’s . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 Applications 173.1 Generals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173.2 Regular 64-Run Designs with 7 ≤ m ≤ 63 Factors . . . . . . . . . . . . . . . . . . 183.3 Comparison with Minimum Aberration Designs . . . . . . . . . . . . . . . . . . . 20

4 Concluding Remarks 25

Bibliography 25

vi

List of Tables

Table 3.1 All possible D1’s and D2’s for designs 224−18 . . . . . . . . . . . . . . . . . 18Table 3.2 (A3, A4, A5)’s of the best designs from our approach with 7 ≤ m ≤ 63 factors 19Table 3.3 (A3, A4, A5)’s of minimum aberration (MA) designs and those of the corre-

sponding best designs from our approach for 7 ≤ m ≤ 32 . . . . . . . . . . 20Table 3.4 (A3, A4, A5) of design D̄ and that of the corresponding MA design D . . . 23Table 3.5 (A3, A4, A5)’s of MA designs and those of the corresponding best designs

from our approach for 33 ≤ m ≤ 63 . . . . . . . . . . . . . . . . . . . . . . 24

vii

List of Figures

Figure 2.1 Road Map for the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

viii

Chapter 1

Introduction

Fractional factorials are widely employed in areas of science and technology owing to theirflexibility and run size economy. The primary problem for us to use these designs is how tochoose an optimal design for a given number of factors and run size. Various criteria have beensuggested to deal with this problem, but it seems impractical to provide optimal designs foreach criterion. The most popular criterion for comparing regular fractional factorials is min-imum aberration, introduced by Fries and Hunter (1980). A generalization of this criterion,called minimum G2-aberration, was proposed by Tang and Deng (1999) for comparing bothregular and nonregular fractional factorials. Although minimum aberration designs are com-pletely known for up to 128 runs, results for more than 128 runs are quite limited. MinimumG2-aberration designs are available for up to 96 runs and in fact only partially available for morethan 32 runs.

This project presents a simple and efficient approach to constructing a “good” large designusing two small minimum G2-aberration designs. We focus on orthogonal factorials with twolevels, which can be classified into two categories, regular fractional factorials and nonregularfractional factorials. A regular fractional factorial 2m−p, having m factors of two levels, m − p

independent factors, and 2m−p runs, is determined by its defining relation which contains p

independent defining words. Regular designs have a property that any two effects are eitherorthogonal or fully aliased. In contrast to a 2m−p design, a nonregular fractional factorial hassome complex aliasing structure, meaning that there exist two partially aliased effects. Tangand Deng (1999) provided a formal definition for both regular and nonregular factorials, whichis introduced in Section 1.1.

The rest of this chapter reviews J-characteristics, orthogonal factorials, and the criteria ofminimum aberration and minimum G2-aberration. Chapter 2 proposes our approach to con-structing a large design using two smaller designs, and then studies some relationships betweenthe large design and the two small designs. The results are presented in Sections 2.2-2.4. ForChapter 3, the general applications of the results derived in Chapter 2 are discussed in Sec-tion 3.1. We then use a specific case to assess the goodness of designs constructed by our

1

approach. We construct regular 64-run factorials with 7 ≤ m ≤ 63 factors using the approachsuggested in Section 2.1. For each number m of factors, we choose a design with the leastaberration from those constructed designs as the best one, and compare it with the minimumaberration design. Chapter 4 summarizes this project and discusses some possible future work.

1.1 Notation and Background

Suppose design D is an orthogonal fractional factorial (regular or nonregular) with m factorsand n runs. For convenience, we write D as a set of m columns, D = {d1, . . . , dm}, or an n×m

matrix, D = (dij), where dij ∈ {−1, 1}. For 1 ≤ k ≤ m and any k-subset u = {dj1 , . . . , djk} of

D, Deng and Tang (1999) defined

Jk(u) = Jk(dj1 , . . . , djk) =

∣∣∣∣∣n∑

i=1dij1 · · · dijk

∣∣∣∣∣ , (1.1)

where dij1 is the ith component of column dj1 . J0(ϕ) = n was also defined. Here, J1(u) = J2(u) =0, as the numbers of the two levels in any column of D are identical, and any two columns areorthogonal in D. Tang and Deng (1999) summarized the Jk(u) values in the following definition.

Definition 1.1.1. The Jk(u) values in (1.1) are called the J-characteristics of design D.

According to (1.1), we have the following fact, which will be used in the proofs of the theo-rems in Sections 2.2-2.4.

Fact 1.1.1. For any k-subset u = {dj1 , . . . , djk} of D and a column d ∈ {d1, . . . , dm} in D, we

have

(a)Jk+1(dj1 , . . . , djk

, In) = Jk(dj1 , . . . , djk),

(b)

dh = d · · · d︸ ︷︷ ︸h

=

In if h is even,

d if h is odd,

(c)

Jh+1(d, . . . , d︸ ︷︷ ︸h

, In) = Jh(d, . . . , d︸ ︷︷ ︸h

) =

n if h is even,

0 if h is odd,

(d)

Jk+h(dj1 , . . . , djk, d, . . . , d︸ ︷︷ ︸

h

) =

Jk(dj1 , . . . , djk) if h is even,

Jk+1(dj1 , . . . , djk, d) if h is odd,

2

where In is the identity column of length n with all 1’s.

Based on the J-characteristics of D, Deng and Tang (1999) introduced the notion of gen-eralized resolution. Let r be the smallest integer such that max|u|=r Jr(u) > 0, where themaximization is over all the subsets of r distinct columns of D. Then the generalized resolutionof design D is defined to be

R(D) = r +[1−max

|u|=rJr(u)/n

]. (1.2)

Clearly, r ≤ R(D) < r + 1. Note that R(D) ≥ 3 for orthogonal designs, as J1(u) = J2(u) = 0.

For a regular design D, Jk(u) = n or 0, as effects in u are either fully aliased or orthogonal.But, for a nonregular factorial, there exists a u such that 0 < Jk(u) < n. According to the valuesof J-characteristics, Tang and Deng (1999) gave a formal definition for regular and nonregulardesigns.

Definition 1.1.2. A fractional factorial D is said to be regular if Jk(u) = n or 0 for all u ⊆ D.It is said to be nonregular if there exists a u ⊆ D such that 0 < Jk(u) < n.

It is clear that the defining relation of a regular design D is the collection of all subsetsu’s such that Jk(u) = n for k = 1, . . . , m. This means that if there exists a k such thatJk(u) = n, then a word in the defining relation is formed by those k columns in u. Let Ak(D)be the number of words of length k in the defining relation, and then the word length patternof design D is defined as the vector, W (D) = (A1(D), A2(D), A3(D), . . . , Am(D)). Obviously,A1(D) = A2(D) = 0.

For two regular factorials D1 and D2 with the same number of factors and run size, theminimum aberration is utilized to compare them. Let r be the smallest integer such thatAr(D1) ̸= Ar(D2). If Ar(D1) < Ar(D2), then D1 is said to have less aberration than D2. If nodesign has less aberration than D1, then we say that D1 has minimum aberration.

The minimum G2-aberration, a generalization of the minimum aberration, is used to assessthe goodness of general fractional factorials. For a design D, let

Bk(D) = 1n2

∑|u|=k

J2k (u) = 1

n2

∑1≤j1<···<jk≤m

J2k (dj1 , . . . , djk

), for 1 ≤ k ≤ m, (1.3)

where B1(D) = B2(D) = 0. For two factorials D1 and D2, let r be the smallest integer such thatBr(D1) ̸= Br(D2). If Br(D1) < Br(D2), then D1 is said to have less G2-aberration than D2.If no design has less G2-aberration than D1, then we say that D1 has minimum G2-aberration.If D is a regular factorial, then Bk(D) = Ak(D), which implies that minimum G2-aberration is

3

equivalent to minimum aberration for regular designs.

For 1 ≤ k ≤ m, Butler (2003) defined

Mk(D) = 1n2

m∑j1=1· · ·

m∑jk=1

J2k (dj1 , . . . , djk

), for 1 ≤ k ≤ m. (1.4)

Clearly, Mk(D) is greater than Bk(D), as it considers all permutations for each collection ofk columns and also allows columns to occur in {dj1 , . . . , djk

} more than once. The quantityMk(D) is instrumental in determining the constants in the theorems of Sections 2.3 and 2.4.

4

Chapter 2

General Theoretical Results

2.1 Constructing A Large Design Using Two Small Designs

We firstly review some basic definitions. Let x = (x1, . . . , xn1)T and y = (y1, . . . , yn2)T . TheKronecker product of two vectors x and y is defined as

x⊗ y = (x1y1, . . . , x1yn2 , . . . , xn1y1, . . . , xn1yn2)T .

Tang (2006) provided a simple way to calculate the J-characteristic of Kronecker products.

Lemma 2.1.1. We have that

J(a1 ⊗ b1, . . . , ak ⊗ bk) = J(a1, . . . , ak)J(b1, . . . , bk),

where aj = (a1j , . . . , an1j)T and bj = (b1j , . . . , bn2j)T for j = 1, . . . , k.

Let D1 and D2 be two factorials, either regular or nonregular. Design D1 is an n1-run designwith m1 factors and can be expressed by a set of m1 columns, D1 = {a1, . . . , am1}. Similarly,D2 has m2 factors and n2 runs, and can be written as a set of m2 columns, D2 = {b1, . . . , bm2}.A large design D can then be obtained by taking the Kronecker product of two designs D1 andD2,

D = D1 ⊗D2

= {a1, . . . , am1} ⊗ {b1, . . . , bm2}

= {a1 ⊗ b1, . . . , a1 ⊗ bm2 , . . . , am1 ⊗ b1, . . . , am1 ⊗ bm2} . (2.1)

Clearly, design D is a factorial with m factors and n runs, where m = m1m2 and n = n1n2.Moreover, D is a regular factorial if both D1 and D2 are regular. Otherwise, D is a nonregularfactorial.

5

For any k-subset u = {ai1 ⊗ bj1 , . . . , aik⊗ bjk

}, where 1 ≤ k ≤ m, by Lemma 2.1.1, we have

Jk(u) = Jk (ai1 ⊗ bj1 , . . . , aik⊗ bjk

)

= Jk (ai1 , . . . , aik) Jk (bj1 , . . . , bjk

) . (2.2)

Here, since each column in D is the Kronecker product of a column from D1 and one from D2 andthe total number of columns in D is m1m2, for any ai ∈ {a1, . . . , am1} and bj ∈ {b1, . . . , bm2},the numbers of ai and bj contributing to D are m1 and m2, respectively. Hence, for an arbitraryk-subset u = {ai1 ⊗ bj1 , . . . , aik

⊗ bjk} with 1 ≤ k ≤ m, each of the collections {ai1 , . . . , aik

} and{bj1 , . . . , bjk

} in (2.2) may contain some repeated columns. This observation is important forthe proofs in the following sections of this chapter.

2.2 Doubling

Doubling is a special case of the construction in (2.1), where

D1 =[1 −11 1

]. (2.3)

Here, design D1 only has two columns a1 = (1, 1)T and a2 = (−1, 1)T . Design D2 is an ordinaryfactorial, either regular or nonregular, with m2 factors and n2 runs. The double of D2 is thendefined as

D = D1 ⊗D2

=[D2 −D2

D2 D2

]. (2.4)

Clearly, design D has m = 2m2 factors and n = 2n2 runs. If D2 is regular, Chen and Cheng(2006) derived a relationship between the word-length pattern of D2 and that of its correspondingD, which is given in Theorem 2.2.1. The same relationship also holds for nonregular designs butit does require a new proof.

Theorem 2.2.1. Suppose k is a positive integer with 1 ≤ k ≤ 2m2. Then

Bk(D) =

0 if k ∈ {1, 2} ,⌊(k−3)/2⌋∑

t=02k−2t−1(m2−(k−2t)

t

)Bk−2t(D2) +

(m2k/2)

if k is a multiple of 4,

⌊(k−3)/2⌋∑t=0

2k−2t−1(m2−(k−2t)t

)Bk−2t(D2) otherwise,

(2.5)

where ⌊x⌋ is the largest integer less than or equal to x.

6

Proof. It is obvious that B1(D) = B2(D) = 0, as J1 = J2 = 0 for any orthogonal design D.For 3 ≤ k ≤ 2m2, since there are only 2 columns in D1, each column in {bj1 , . . . , bjk

} occursat most twice when we calculate Jk(u) in ( 2.2). Let t be the number of columns occurringexactly twice in {bj1 , . . . , bjk

}. We have that the remaining (k − 2t) columns in {bj1 , . . . , bjk},

different from the above t columns, are all distinct. This implies that there are(m2−(k−2t)

t

)ways

to choose the repeating columns for each {bj1 , . . . , bjk}. By part (d) of Fact 1.1.1, we then have

that Jk (bj1 , . . . , bjk) =

(m2−(k−2t)t

)Jk−2t

(bj1 , . . . , bjk−2t

). On the other hand, design D1 in (2.3)

gives Jk (ai1 , . . . , aik) = 0 or 2, and Jk (ai1 , . . . , aik

) = 2, only if the number of a2 = (−1, 1)T

in {ai1 , . . . , aik} is even. As there are t columns occurring twice in {bj1 , . . . , bjk

}, column a2

is included in {ai1 , . . . , aik} at least t times. We then study two situations, (a) t is even and

(b) t is odd, although the final results are the same for both situations. For situation (a),since t is even, the number of a2’s which correspond to k − 2t distinct columns in {bj1 , . . . , bjk

}is even for otherwise Jk(u) = 0. Then the total number of ways to choose the number ofa2’s is

(k−2t0)

+(k−2t

2)

+(k−2t

4)

+ · · · = 2k−2t/2. Resembling situation (a), the number of a2’swhich correspond to k − 2t distinct columns is odd, as t is odd in situation (b). Then there are(k−2t

1)+(k−2t

3)+(k−2t

5)+ · · · = 2k−2t/2 ways to choose the number of a2’s. As J1 = J2 = 0 for any

orthogonal design D, t must satisfy k−2t ≥ 3 in order to have Jk−2t

(bj1 , . . . , bjk−2t

)> 0. Since t

is a nonnegative integer, we have t ∈ [0, (k − 3)/2], if k is odd, and t ∈ [0, (k − 4)/2], if k is even.We have two cases: (i) k is odd, and (ii) k is even. For any k-subset u = {ai1 ⊗ bj1 , . . . , aik

⊗ bjk},

where 3 ≤ k ≤ 2m2, case (i) gives

Bk(D) = 1(2n2)2

∑|u|=k

J2k (u)

= 1(2n2)2

∑|u|=k

J2k (ai1 , . . . , aik

) J2k (bj1 , . . . , bjk

)

=(k−3)/2∑

t=02k−2t/2

(m2 − (k − 2t)

t

)1n2

2

∑1≤j1<···<jk−2t≤m2

J2k−2t

(bj1 , . . . , bjk−2t

)

=(k−3)/2∑

t=02k−2t/2

(m2 − (k − 2t)

t

)Bk−2t(D2), (2.6)

For case (ii), except for a similar term like (2.6), Bk(D) contains an extra constant term for aneven k/2, as there exists t = k/2 under (ii) such that k−2t = 0. This implies that there are k/2distinct columns in {bj1 , . . . , bjk

}, each column occurring twice. We then have that the numberof a2’s in {ai1 , . . . , aik

} is exact k/2. Since Jk (bj1 , . . . , bjk) = n2 by part (c) of Fact 1.1.1, and

Jk (ai1 , . . . , aik) = n1 = 2 > 0 only when k/2 is even, a non-zero constant term exists only when

k/2 is even. The number of ways to choose those k/2 columns is(m2

k/2). The final result for an

even k/2 is

Bk(D) =(k−4)/2∑

t=02k−2t/2

(m2 − (k − 2t)

t

)Bk−2t(D2) +

(m2k/2

). (2.7)

Combining (2.6) and (2.7), Theorem 2.2.1 is obtained.

7

Using Theorem 2.2.1, detailed expressions for B3(D), B4(D) and B5(D) are easily obtained,which are shown below

B3(D) = 4B3(D2), (2.8)

B4(D) = 8B4(D2) +(

m22

), (2.9)

and

B5(D) = 16B5(D2) + 4(m2 − 3)B3(D2). (2.10)

These equations will be used in the next chapter.

2.3 Small Design Without A Column of 1’s

Design D in Section 2.2 has a restriction in that it can only double the number m2 of factorsand the run size n2 of the original design D2. This implies that the number m of factors andthe run size n of D are fixed for given m2 and n2 of D2. To make m and n more flexible, weconstruct D by using a general design D1. Design D1 discussed in this section is any factorial,with the only requirement that it does not contain a column of all 1’s. Note that we alwaysassume D2 does not have a column of all 1’s.

As D1 is relaxed, Bk(D) is also related to Bs(D1) for s ≤ k. A relationship can be establishedamong Bs(D1), Bt(D2) and Bk(D) for s, t ≤ k, which is shown in the following theorem.

Theorem 2.3.1. Suppose k is a positive integer with 1 ≤ k ≤ m1m2. Then

Bk(D) =

0 if k ∈ {1, 2} ,l∑

s=0

l∑t=0

CstBk−2s(D1)Bk−2t(D2) if k ≥ 3 is odd,

l∑s=0

l∑t=0

CstBk−2s(D1)Bk−2t(D2) +l∑

s=0C

(1)s Bk−2s(D1) +

l∑t=0

C(2)t Bk−2t(D2) + C if k ≥ 3 is even,

(2.11)

where l = ⌊(k − 3)/2⌋, and C00 > 0. All constants in (2.11) depend on m1, m2, n1, n2, not onchoices of D1 and D2 for given m1, m2, n1, n2.

Bk(D1) and Bk(D2) ←→ Bk(D)xy Ixy III

Mk(D1) and Mk(D2) II←→ Mk(D)

Figure 2.1: Road Map for the Proof

8

Proof. Obviously, B1(D) = B2(D) = 0. For 3 ≤ k ≤ m1m2, instead of directly studying arelationship of Bk’s among designs D1, D2 and D, we separate it into three steps I, II, andIII, which is shown in Figure 2.1. Step I connects Bk(D1) and Bk(D2) with Mk(D1) andMk(D2), respectively. Step II links Mk(D1) and Mk(D2) with Mk(D). Since Mk considers allpermutations for each collection of k columns and allows the columns occurring more than oncein each collection, the relationship in Step II is straightforward and given by

Mk(D) = Mk(D1)Mk(D2), for 1 ≤ k ≤ m1m2. (2.12)

The last step gives a connection between Mk(D) and Bk(D). In practice, two relationships inStep I are identical and the inverse of this relationship is what Step III needs, as D1, D2 andD are general factorials without a column of 1’s. Hence, we choose design D2 to study thisrelationship. Let t∗ be the number of columns occurring more than once in {bj1 , . . . , bjk

}, saycolumns bjk−t∗+1 , . . . , bjk

, and hg be the number of bjg ’s in {bj1 , . . . , bjk} for k − t∗ + 1 ≤ g ≤ k.

We then consider two situations, (i) all hg’s are even, and (ii) at least one hg is odd. Situation (i)gives that there exists a t such that

∑kg=k−t∗+1 hg = 2t. We then have b

hk−t∗+1jk−t∗+1

· · · bhkjk

= In2 bypart (d) of Fact 1.1.1, which implies that Jk (bj1 , . . . , bjk

) = Jk−2t

(bj1 , . . . , bjk−2t

). For situation

(ii), suppose hk−t∗+1, . . . , hg′ are odd, where g′ can be any number in {k − t∗ + 1, . . . , k}. Thenthere exists a t such that

∑g′

g=k−t∗+1 (hg − 1) +∑k

g′ hg =∑k

g=k−t∗+1 hg − (g′ − k + t∗) = 2t. Bypart (b) of Fact 1.1.1, we then obtain b

hk−t∗+1jk−t∗+1

· · · bhkjk

= bjk−t∗+1 · · · bjg′ , and Jk (bj1 , . . . , bjk) =

Jk−2t

(bj1 , . . . , bj

k−∑

hg, bjk−t∗+1 , . . . , bjg′

)= Jk−2t

(bj1 , . . . , bjk−2t

). Both situations (i) and (ii)

give that Jk are related to Jk−2t’s with 3 ≤ k − 2t ≤ k. We consider two cases. Case (1): k isodd. We then have

Mk(D2) = 1n2

2

m∑j1=1· · ·

m∑jk=1

J2k (bj1 , . . . , bjk

)

=(k−3)/2∑

t=0C

∗(2)t

∑1≤j1<···<jk−2t≤m2

1n2

2J2

k−2t

(bj1 , . . . , bjk−2t

)

=(k−3)/2∑

t=0C

∗(2)t Bk−2t(D2), (2.13)

where C∗(2)t is a positive constant, as C

∗(2)t is the product of two numbers, the number of ways

to choose the repeating columns and that of ways to permute k columns in each collection.Similarly, for D1 and D, we have

Mk(D1) =(k−3)/2∑

s=0C∗(1)

s Bk−2s(D1), (2.14)

Mk(D) =(k−3)/2∑

v=0C∗(3)

v Bk−2v(D), (2.15)

9

where C∗(1)s and C

∗(3)v are positive constants for 0 ≤ s, v ≤ (k − 3)/2. Using (2.12), a recursion

formula for an odd k is obtained

(k−3)/2∑v=0

C∗(3)v Bk−2v(D) =

(k−3)/2∑s=0

C∗(1)s Bk−2s(D1)

(k−3)/2∑t=0

C∗(2)t Bk−2t(D2)

=(k−3)/2∑

s=0

(k−3)/2∑t=0

C∗stBk−2s(D1)Bk−2t(D2). (2.16)

A formula for Bk(D) can then be obtained from the above recursion formula,

Bk(D) =(k−3)/2∑

s=0

(k−3)/2∑t=0

CstBk−2s(D1)Bk−2t(D2), (2.17)

where C00 is a positive coefficient for the leading term Bk(D1)Bk(D2) for 3 ≤ k ≤ m1m2, as canbe verified by induction. For k = 3, using (2.16), it is obvious that B3(D) = C∗

00C

∗(3)0

B3(D1)B3(D2)

with C∗00

C∗(3)0

> 0. For k > 3, let k∗ be the maximum odd number in the range [3, m1m2]. Supposethat (2.17) with a positive C00 is true for k ≤ (k∗ − 2). For k = k∗, using (2.16), we have

C∗(3)0 Bk∗(D) =

(k∗−3)/2∑s=0

(k∗−3)/2∑t=0

C∗stBk∗−2s(D1)Bk∗−2t(D2)−

(k∗−3)/2∑v=1

C∗(3)v Bk∗−2v(D)

= C∗00Bk∗(D1)Bk∗(D2)

+(k∗−3)/2∑

t=0C∗

0tBk∗(D1)Bk∗−2t(D2) +(k∗−3)/2∑

s=0C∗

s0Bk∗−2s(D1)Bk∗(D2)

+(k∗−3)/2∑

s=1

(k∗−3)/2∑t=1

C∗stBk∗−2s(D1)Bk∗−2t(D2)−

(k∗−3)/2∑v=1

C∗(3)v Bk∗−2v(D)

= C∗00Bk∗(D1)Bk∗(D2) +

(k∗−3)/2∑t=0

C∗0tBk∗(D1)Bk∗−2t(D2)

+(k∗−3)/2∑

s=0C∗

s0Bk∗−2s(D1)Bk∗(D2) +(k∗−3)/2∑

s=1

(k∗−3)/2∑t=1

C∗stBk∗−2s(D1)Bk∗−2t(D2)

−(k∗−3)/2∑

v=1C∗(3)

v

(k∗−2v−3)/2∑s=0

(k∗−2v−3)/2∑t=0

C∗stBk∗−2v−2s(D1)Bk∗−2v−2t(D2)

= C∗00Bk∗(D1)Bk∗(D2) +

(k∗−3)/2∑t=0

C∗0tBk∗(D1)Bk∗−2t(D2)

+(k∗−3)/2∑

s=0C∗

s0Bk∗−2s(D1)Bk∗(D2) +(k∗−3)/2∑

s=1

(k∗−3)/2∑t=1

C∗∗st Bk∗−2s(D1)Bk∗−2t(D2)

=(k∗−3)/2∑

s=0

(k∗−3)/2∑t=0

C∗∗st Bk∗−2s(D1)Bk∗−2t(D2), (2.18)

10

where C∗∗00 = C∗

00 > 0. As the coefficient C∗(3)0 of Bk(D) is positive, equation (2.17) is true for

3 ≤ k ≤ k∗. Case (2): k is even. There exist some t∗’s such that∑k

g=k−t∗+1 hg = 2t = k, whichimplies that Jk (bj1 , . . . , bjk

) = n2, only if all hg’s are even. Let m∗ be the number of values oft∗, we then obtain

Mk(D2) =(k−4)/2∑

t=0C

∗(2)t Bk−2t(D2) +

m∗∑t∗=1

C ′2t∗ ,

=(k−4)/2∑

t=0C

∗(2)t Bk−2t(D2) + C ′

2, (2.19)

where C ′2t∗ is the number of ways to choose the t∗ columns. Similar to the case of odd k, Mk(D1)

and Mk(D) can be easily written as

Mk(D1) =(k−4)/2∑

s=0C∗(1)

s Bk−2s(D1) + C ′1, (2.20)

Mk(D) =(k−4)/2∑

v=0C∗(3)

v Bk−2v(D) + C ′, (2.21)

where C∗(1)s and C

∗(3)v are positive constants. A recursive formula is derived using (2.12),

(k−4)/2∑v=0

C∗(3)v Bk−2v(D) + C ′ =

(k−4)/2∑s=0

C∗(1)s Bk−2s(D1) + C ′

1

(k−4)/2∑t=0

C∗(2)t Bk−2t(D2) + C ′

2

.

By induction, the formula for each Bk(D) with an even k is

Bk(D) =k−4

2∑s=0

k−42∑

t=0CstBk−2s(D1)Bk−2t(D2)+

k−42∑

s=0C(1)

s Bk−2s(D1)+k−4

2∑t=0

C(2)t Bk−2t(D2)+C, (2.22)

where C00 > 0. We then arrive at (2.11) with a positive coefficient of the leading termBk(D1)Bk(D2) for 3 ≤ k ≤ m1m2.

Obviously, it is hard to determine the constants in (2.11) explicitly for all s and t. Here,we only focus on the cases of k = 3, 4, 5, as they are useful in Chapter 3. The simplest case isk = 3. It is evident that

M3(D1) = 3!B3(D1),

as M3 considers the permutations of columns in {ai1 , ai2 , ai3}. Similarly, we have

M3(D2) = 3!B3(D2), (2.23)

and

M3(D) = 3!B3(D). (2.24)

11

Using (2.12) we obtainB3(D) = 3!B3(D1)B3(D2). (2.25)

For k = 4, by equation (2.20), we have

M4(D1) = C∗(1)0 B4(D1) + C ′

1

= 4!B4(D1) +((

42

)(m12

)+(

m11

))

= 4!B4(D1) + 6(

m12

)+ m1, (2.26)

where 4! is the number of ways to permute four distinct columns in {ai1 , ai2 , ai3 , ai4},(4

2)(m1

2)

is the product of two numbers, the number of ways to choose two distinct columns and that topermute four columns, and

(m11)

is the number of ways to choose one column. Similarly for D2

and D, we obtain

M4(D2) = 4!B4(D2) + 6(

m22

)+ m2, (2.27)

and

M4(D) = 4!B4(D) + 6(

m

2

)+ m. (2.28)

Again using (2.12) and after some algebraic calculation, we derive that

B4(D) =4!B4(D1)B4(D2) +(

m1 + 6(

m12

))B4(D2)

+(

m2 + 6(

m22

))B4(D1) +

(m12

)(m22

). (2.29)

Clearly, using equation (2.14) with k = 5, we get

M5(D1) = C∗(1)0 B5(D1) + C

∗(1)1 B3(D1) = 5!B5(D1) + C

∗(1)1 B3(D1),

where 5! is the number of the ways to permute five distinct columns in each subset. For thesecond term in the above equation, it is obvious that there must be only one column occurringat least twice, say columns ai4 = ai5 . Two situations are possible: (i) there exists a h such thatai4 = aih

, where h = 1, 2, 3, and (ii) ai4 ̸= aihfor h = 1, 2, 3. For situation (i) , there are

(31)

ways to choose a column from {ai1 , ai2 , ai3}, and(5

3)(2

1)

ways to permute those five columns.Situation (ii) has

(m1−31)

ways to choose a column from the set without columns ai1 , ai2 , andai3 , and has

(52)3! ways to permute those five columns. We thus have

12

M5(D1) = 5!B5(D1) +((

31

)(53

)(21

)+(

m1 − 31

)(52

)3!)

B3(D1)

= 5!B5(D1) + 60(m1 − 2)B3(D1). (2.30)

Obviously, M5(D2) and M5(D) are

M5(D2) = 5!B5(D2) + 60(m2 − 2)B3(D2), (2.31)

and

M5(D) = 5!B5(D) + 60(m− 2)B3(D). (2.32)

Finally, some simple algebra leads to

B5(D) = 5!B5(D1)B5(D2) + 60(m2 − 2)B5(D1)B3(D2) + 60(m1 − 2)B3(D1)B5(D2)

+ (27m1m2 − 60m1 − 60m2 + 126)B3(D1)B3(D2). (2.33)

2.4 Small Design with A Column of 1’s

Design D1 discussed in this section contains one column of 1’s which is the identity columnIn1 . This implies that design D constructed by such a D1 has extra m2 columns. We also derivea similar relationship among Bs(D1), Bt(D2), and Bk(D) for s, t ≤ k, which is shown in thefollowing theorem.

Theorem 2.4.1. Suppose k is a positive integer with 1 ≤ k ≤ m1m2. Then

Bk(D) =

0 if k ∈ {1, 2} ,k−3∑s=0

l∑t=0

CstBk−s(D1)Bk−2t(D2) +l∑

t=0C

(2)t Bk−2t(D2) if k ≥ 3 is odd,

k−3∑s=0

l∑t=0

CstBk−s(D1)Bk−2t(D2) +l∑

t=0C

(2)t Bk−2t(D2) +

k−3∑s=0

C(1)s Bk−s(D1) + C if k ≥ 3 is even,

(2.34)

where l = ⌊(k − 3)/2⌋, and C00 > 0. All constants in (2.34) depend on m1, m2, n1, n2, not onchoices of D1 and D2 for given m1, m2, n1, n2.

Proof. This proof is similar to that of Theorem 2.3.1. As D1 contains column In1 , Mk(D1)depends on all Bk−s(D1) for 0 ≤ s ≤ k − 3. A constant term always exists, regardless of theparity of k, because of a special collection with length k, {In1 , . . . , In1}. We then have

Mk(D1) =(k−3)∑s=0

C∗(1)s Bk−s(D1) + C∗, (2.35)

13

where C∗(1)s is a positive constant, as it is the product of two numbers, the number of ways

to choosing the repeating columns and that of ways to permute columns in the correspondingcollection. Since D2 and D are two ordinary orthogonal designs, the expressions for Mk derivedin the proof of Theorem 2.3.1 are still valid. Mk(D2) and Mk(D) with an odd k are shown in(2.13) and (2.15) with positive coefficients, respectively. For an even k, equations (2.19) and(2.21) with positive coefficients are also the expressions for Mk(D2) and Mk(D) in this section,respectively. Using (2.12), recursive formulas are

(k−3)/2∑v=0

C∗(3)v Bk−2v(D) =

k−3∑s=0

C∗(1)s Bk−s(D1) + C∗

(k−3)/2∑t=0

C∗(2)t Bk−2t(D2) if k is odd,

(k−4)/2∑v=0

C∗(3)v Bk−2v(D) + C ′ =

k−3∑s=0

C∗(1)s Bk−s(D1) + C∗

(k−4)/2∑t=0

C∗(2)t Bk−2t(D2) + C ′

2

if k is even.

Using induction for odd k’s and even k’s respectively, we obtain

Bk(D) =

k−3∑s=0

(k−3)/2∑t=0

CstBk−s(D1)Bk−2t(D2) +(k−3)/2∑

t=0C

(2)t Bk−2t(D2) if k is odd,

k−3∑s=0

(k−4)/2∑t=0

CstBk−s(D1)Bk−2t(D2) +(k−4)/2∑

t=0C

(2)t Bk−2t(D2) +

k−3∑s=0

C(1)s Bk−s(D1) + C if k is even,

where C00 > 0 for both odd k’s and even k’s.

Here, we also determine the constants in Bk(D) using Mk of the corresponding designs D1,D2 and D for k = 3, 4, 5. Clearly, Mk(D2) and Mk(D) for k = 3, 4, 5 were given in Section 2.3,as D2 and D are factorials without one column of 1’s. For design D1 with one column of 1’s,Mk(D1) with k = 3, 4, 5 are derived as follows.

For k = 3, using (2.35), we have

M3(D1) = C∗(1)0 B3(D1) + C∗,

where C∗(1)0 = 3!, as the number of ways to permute three distinct columns in each subset is

3!. The term C∗ considers the collection {ai1 , ai2 , ai3} with repeating columns. As there areonly three columns in each collection, two situations should be studied, (i) a column occurstwice, and (ii) a column occurs 3 times. For situation (i), J2

3 (ai1 , ai2 , ai3) > 0, only when eachcollection {ai1 , ai2 , ai3} contains one column In1 apart from two identical columns. There are(m1− 1) ways to choose the identical column and

(31)

ways to permute those three columns. Forsituation (ii), J2

3 (ai1 , ai2 , ai3) > 0, only when each collection {ai1 , ai2 , ai3} contains three In1 ’s.

14

We then obtain

M3(D1) = 3!B3(D1) +((

31

)(m1 − 1) + 1

)= 3!B3(D1) + 3(m1 − 1) + 1.

Combining this equation with (2.23) and (2.24), we have

B3(D) = 3!B3(D1)B3(D2) + (3m1 − 2)B3(D2). (2.36)

For k = 4, we obtain

M4(D1) = C∗(1)0 B4(D1) + C

∗(1)1 B3(D1) + C∗.

Obviously, C∗(1)0 = 4!. As each collection of B3(D1) contains one column In1 which is different

from the other three distinct columns, there are 4! ways to permute those four columns. The lastterm is from the collections which have the repeating columns. Let h be the number of columnsoccurring more than once. Clearly, h = 1, 2. For h = 1, J2

4 (ai1 , . . . , ai4) > 0, only when ai1 = aig

for g = 2, 3, 4. There are m1 distinct columns which can be used to form such collections. Forh = 2, since there are four columns in each collection, each column occurs exactly twice. thenumber of ways to choose two distinct columns is

(m12). There are

(42)

ways to permute columnsin each collection. We thus obtain

M4(D1) = 4!B4(D1) + 4!B3(D1) +(

m1 +(

m12

)(42

))

= 4!B4(D1) + 4!B3(D1) +(

m1 + 6(

m12

)).

Using (2.27) and (2.28), B4(D) has the following expression:

B4(D) = 4!B4(D1)B4(D2) + 4!B3(D1)B4(D2) +(

m1 + 6(

m12

))B4(D2)

+(

m2 + 6(

m22

))(B4(D1) + B3(D1)) +

(m12

)(m22

). (2.37)

For k = 5, it is obvious that

M5(D1) = C∗(1)0 B5(D1) + C

∗(1)1 B4(D1) + C

∗(1)2 B3(D1) + C∗.

Clearly, C∗(1)0 = C

∗(1)1 = 5!. The term C

∗(1)2 B3(D1) is identical to the second term 60(m1 −

2)B3(D1) in (2.30). The interpretation of the coefficient of B3(D1) was also given in Section 2.3.For the constant term, we let h be the number of In1 ’s. Since there are five columns in acollection, h is odd, clearly h = 1, 3, 5. For h = 1 there are two cases, (a) the remaining fourcolumns are identical, (b) not all of them are the same. Case (a) gives (m1−1) ways to choose a

15

repeating column and(5

4)

ways to permute those five columns in each collection. Case (b) showsthat J2

5 (ai1 , . . . , ai5) > 0, only when two distinct columns occur exact twice. There are(m1−1

2)

ways to choose two distinct columns and(5

2)(3

2)

ways to permute those five columns. For h = 3,in order to obtain J2

5 (ai1 , . . . , ai5) > 0, the left two columns in the collection must be the same.The number of ways to choose a column is

(m1−11)

and there are(5

2)

ways to permute those fivecolumns. It is obvious that only J2

5 (In1 , . . . , In1) > 0 for h = 5. M5(D1) can then be obtained

M5(D1) = 5!B5(D1) + 5!B4(D1) + 60(m1 − 2)B3(D1)

+((

m1 − 11

)(54

)+(

m1 − 12

)(52

)(32

)+(

m1 − 11

)(52

)+ 1

)

= 5!B5(D1) + 5!B4(D1) + 60(m1 − 2)B3(D1) + 30(

m1 − 12

)+ 15m1 − 14

Combining the above with (2.31) and (2.32), we get that

B5(D) = 5!B5(D1)B5(D2) + 5!B4(D1)B5(D2)

+ 60(m2 − 2) (B5(D1) + B4(D1)) B3(D2)

+ 60(m1 − 2)B3(D1)B5(D2) +(

30(

m1 − 12

)+ 15m1 − 14

)B5(D2)

+ (27m1m2 − 60m1 − 60m2 + 126)B3(D1)B3(D2)

+ (6m21m2 − 15m2

1 − 14m1m2 + 33m1 + 8m2 − 18)B3(D2). (2.38)

These three quantities B3(D), B4(D) and B5(D) play an important role in the next chapter.

16

Chapter 3

Applications

3.1 Generals

Section 2.1 presents a simple approach to constructing a large design using two small designs.But, it may be computationally difficult or impractical to assess the goodness of this large designusing minimum G2-aberration by directly evaluating values of the associated Bk for 1 ≤ k ≤ m.The properties derived from Sections 2.2-2.4 provide us with a simple way to achieve this.

Theorem 2.2.1 shows that Bk(D) of the doubled design D is a linear combination of Bk(D2),Bk−2(D2), . . . of the original design D2, with a positive coefficient for the leading term Bk(D2).Theorem 2.3.1 says that Bk(D) of design D is a linear combination of Bk(D1)Bk(D2), Bk(D1)Bk−2(D2), . . . of the original designs D1 and D2, with a positive coefficient for the leading termBk(D1)Bk(D2). Resembling Theorem 2.3.1, Theorem 2.4.1 states that Bk(D) is a linear combi-nation of Bk(D1)Bk(D2), Bk(D1)Bk−2(D2), . . . of the original designs D1 and D2, with a positivecoefficient for the leading term Bk(D1)Bk(D2). These results imply that if we choose D1 andD2 with small Bk’s, the corresponding D also has small Bk(D)’s. A design with small Bk’s canthen be obtained using two small minimum G2-aberration designs.

Specific expressions for the B3(D) value in Theorems 2.2.1, 2.3.1, and 2.4.1 are given in (2.8),(2.25), and (2.36), respectively. The corresponding B4(D)’s are presented in (2.9), (2.29), and(2.37). We note that in each case, B3(D) depends on B3(D2). However, each B4(D) contains aconstant term. If we choose a design D2 with B3(D2) = 0, then we obtain the corresponding D

with B3(D) = 0, which means that our approach can construct a large design with 4 ≤ R(D) < 5using a smaller design D2 with R(D2) ≥ 4, where R(D) and R(D2) are the generalized reso-lutions of D and D2, respectively. We note that R(D2) ≥ 4 requires some restriction on thenumber of factors and the run size. It is well known that design D2 with R(D2) ≥ 4 exists, onlyif m2 ≤ n2/2. This implies that given a run size n, design D must have 4 ≤ R(D) < 5, if thenumber of factors is small. On the other hand, D has 3 ≤ R(D) < 4, if the number of factors islarge. In most cases, B3(D), B4(D), and B5(D) are enough to compare D’s for a given m and n.

17

In the next two sections, we use regular designs with 7 ≤ m ≤ 63 factors and 64 runs toillustrate our approach.

3.2 Regular 64-Run Designs with 7 ≤ m ≤ 63 Factors

In order to evaluate our approach, the best design from our approach must be found for eachm value. According to the value of m, we consider two situations, (i) m is a composite number;and (ii) m is a prime.

Designs under situation (i) can be constructed by our approach directly. For each compositenumber m, we consider all possible combinations of D1 and D2 that satisfy m1m2 = m andn1n2 = 64. For each combination, we use the corresponding minimum aberration designs D1

and D2. Based on the type of D1, three components (A3(D), A4(D), A5(D)) of the word lengthpattern can be computed using the associated results in Sections 2.2-2.4. We then compareall the resulting designs by the minimum aberration criterion, and the design with the leastaberration is the best design from our approach. This best design is then compared with thecorresponding minimum aberration design in Section 3.3.

For instance, for m = 24, all the possible small designs, D1 and D2 that satisfy m1m2 = 24and n1n2 = 64, are given in Table 3.1, where D(i) is the saturated design of i independentfactors with a column of 1’s added. This table also contains A3, A4, A5 values of the minimumaberration designs D1 and D2 in each combination, respectively. The corresponding (A3, A4, A5)of design D is computed and given in the last column of Table 3.1. Obviously, design D with(A3(D), A4(D), A5(D)) = (0, 370, 0) has the least aberration, and is the best design for m = 24from our approach.

Table 3.1: All possible D1’s and D2’s for designs 224−18

Number of Factors Runs D D1 D2 Dm1 m2 n1 n2 D1 ⊗D2 (A3, A4, A5) (A3, A4, A5) (A3, A4, A5)

2 12 4 16 22 ⊗ 212−8 (0, 0, 0) (16, 39, 48) (0, 378, 0)

2 32[1 −11 1

]⊗ 212−7 (0, 0, 0) (0, 38, 0) (0, 370, 0)

4 6 8 8 24−1 ⊗ 26−3 (0, 1, 0) (4, 3, 0) (0, 378, 0)4 16 D(2)⊗ 26−2 (1, 0, 0) (0, 30, 0) (0, 378, 0)

8 3 8 8 D(3)⊗ 23 (7, 7, 0) (0, 0, 0) (0, 378, 0)16 4 28−4 ⊗ 23−1 (0, 14, 0) (1, 0, 0) (0, 378, 0)

For prime values of m, our approach does not apply directly. In this case, a design witha prime number m of factors can be obtained by the method of deleting one column from adesign with m + 1 factors, which can be constructed by our approach. It can be done as follows.

18

Each time we delete one column from the best design constructed with m + 1 factors, we thencompute A3, A4, A5 values of the design containing the remaining columns. Obviously, the totalnumber of designs with m factors is m + 1. The best design with m factors having the leastaberration is then found by comparing all those m + 1 designs.

Table 3.2: (A3, A4, A5)’s of the best designs from our approach with 7 ≤ m ≤ 63 factors

Number of Factors (A3, A4, A5) Number of Factors (A3, A4, A5)7∗ (0, 3, 0) 36 (64, 1337, 4544)8 (0, 6, 0) 37∗ (80, 1401, 5760)9 (0, 9, 0) 38 (96, 1483, 7040)10 (0, 10, 0) 39∗ (112, 1579, 8400)11∗ (0, 10, 0) 40 (128, 1694, 9856)12 (0, 15, 0) 41∗ (144, 1822, 11432)13∗ (0, 19, 24) 42 (160, 1970, 13136)14 (0, 29, 32) 43∗ (176, 2146, 14960)15 (0, 30, 60) 44 (192, 2335, 16960)16 (0, 52, 64) 45 (210, 2520, 19215)17∗ (0, 64, 96) 46 (224, 2773, 21504)18 (0, 84, 128) 47∗ (240, 3025, 24080)19∗ (0, 100, 192) 48 (256, 3300, 26880)20 (0, 125, 256) 49 (294, 3479, 29841)21 (0, 210, 0) 50 (304, 3836, 33184)22 (0, 255, 0) 51∗ (328, 4140, 36744)23∗ (0, 307, 0) 52 (352, 4469, 40608)24 (0, 370, 0) 53∗ (376, 4821, 44800)25∗ (0, 438, 0) 54 (400, 5199, 49344)26 (0, 518, 0) 55∗ (424, 5603, 54264)27∗ (0, 606, 0) 56 (448, 6034, 59584)28 (0, 707, 0) 57∗ (476, 6482, 65240)29∗ (0, 819, 0) 58 (504, 6958, 71344)30 (0, 945, 0) 59∗ (532, 7462, 77924)31∗ (0, 1085, 0) 60 (560, 7995, 85008)32 (0, 1240, 0) 61∗ (590, 8555, 92568)33∗ (16, 1240, 1120) 62 (620, 9145, 100688)34 (32, 1256, 2240) 63 (651, 9765, 109368)35∗ (48, 1288, 3376)

This delete-one-column method is in fact quite versatile. It can also be applied when m

is a composite number, and can sometimes produce better designs than the direct method.Table 3.2 provides three components (A3(D), A4(D), A5(D)) of the word length pattern of thebest design for each 7 ≤ m ≤ 63 found by our method. The entries with a "∗" identify thosecases where the designs are found using the delete-one-method. Noteworthy are the entries form = 25, 27, 33, 35, 39, 51, 55, and 57. Although these m values are composite, the best designsgiven in the table are actually obtained using the method of deleting one column.

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3.3 Comparison with Minimum Aberration Designs

To assess the goodness of the best designs constructed from our approach, we compare themwith the corresponding minimum aberration designs. We separately consider two ranges of m

values, 7 ≤ m ≤ 32 and 33 ≤ m ≤ 63. The complete catalogue for regular 64-run designs with7 ≤ m ≤ 32 factors of resolution 4 is available in Chen, Sun, and Wu (1993) and Xu (2009).For designs with 33 ≤ m ≤ 63 factors, there is no summary available for the correspondingminimum aberration designs. Hence, we compute A3, A4, A5 values for the minimum aberrationdesigns with 33 ≤ m ≤ 63 factors, and then make a comparison.

The complete catalogue for 7 ≤ m ≤ 32 includes three components (A3, A4, A5) of the wordlength pattern for each nonisomorphic design with m factors, which is given in Table 3.3. The(A3, A4, A5) values of the best designs from our approach are also given in Table 3.3. From thistable, we see that (A3, A4, A5)’s of designs with m = 15, 19, 20 from our approach are identicalto those of the corresponding minimum aberration designs. For m = 30, 31, and 32, the designsconstructed by our approach are equivalent to the minimum aberration designs, which is becausethese designs are unique. Generally, as the m value increases, the result of (A3, A4, A5) derivedfrom our approach is getting closer to that from the minimum aberration design.

Table 3.3: (A3, A4, A5)’s of minimum aberration (MA) designs and those of the correspondingbest designs from our approach for 7 ≤ m ≤ 32

Number of Factors MA designs Constructed designs Number of Factors MA designs Constructed designs(m) (A3, A4, A5) (A3, A4, A5) (m) (A3, A4, A5) (A3, A4, A5)

7 (0, 0, 0) (0, 3, 0) 20 (0, 125, 256) (0, 125, 256)8 (0, 0, 2) (0, 6, 0) 21 (0, 204, 0) (0, 210, 0)9 (0, 1, 4) (0, 9, 0) 22 (0, 250, 0) (0, 255, 0)10 (0, 2, 8) (0, 10, 0) 23 (0, 304, 0) (0, 307, 0)11 (0, 4, 14) (0, 10, 0) 24 (0, 365, 0) (0, 370, 0)12 (0, 6, 24) (0, 15, 0) 25 (0, 435, 0) (0, 438, 0)13 (0, 14, 28) (0, 19, 24) 26 (0, 515, 0) (0, 518, 0)14 (0, 22, 40) (0, 29, 32) 27 (0, 605, 0) (0, 606, 0)15 (0, 30, 60) (0, 30, 60) 28 (0, 706, 0) (0, 707, 0)16 (0, 43, 81) (0, 52, 64) 29 (0, 819, 0) (0, 819, 0)17 (0, 59, 108) (0, 64, 96) 30 (0, 945, 0) (0, 945, 0)18 (0, 78, 144) (0, 84, 128) 31 (0, 1085, 0) (0, 1085, 0)19 (0, 100, 192) (0, 100, 192) 32 (0, 1240, 0) (0, 1240, 0)

To make a comparison for designs with 33 ≤ m ≤ 63 factors, we firstly deduce (A3, A4, A5)’sof minimum aberration designs using the complementary designs. Let H be the saturated designof 64 runs for 63 factors, and D ⊆ H be a design of m factors. The corresponding complementarydesign with m̄ factors is denoted by D̄ = H \D, where m̄ = 63−m. Tang and Wu (1996) proved

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that

A3(D) = constant−A3(D̄), (3.1)

A4(D) = constant + A3(D̄) + A4(D̄), (3.2)

and

A5(D) = constant− (2f−1 −m)A3(D̄)−A4(D̄)−A5(D̄), (3.3)

where f is the number of independent factors. In our case, f = 6. The above equations give usa method of computing A3(D), A4(D), and A5(D) from A3(D̄), A4(D̄), and A5(D̄). This raisesthe question of how to find the values of A3(D̄), A4(D̄), and A5(D̄), if design D has minimumaberration. Tang and Wu (1996) also suggested a rule for identifying the minimum aberrationdesigns.

Rule 3.3.1. A design D∗ has minimum aberration if:

(i) A3(D̄∗) = max A3(D̄) over all∣∣∣D̄∣∣∣ = m̄,

(ii) A4(D̄∗) = min{

A4(D̄) : A3(D̄) = A3(D̄∗)}

;

(iii) A5(D̄∗) = max{

A5(D̄) : A3(D̄) = A3(D̄∗) and A4(D̄) = A4(D̄∗)}

and

(iv) D̄∗ is the unique set (up to isomorphism) satisfying (iii).

It is obvious that, first of all, we should find 64-run design D̄ with the maximum value ofA3. To obtain such designs, we use the following fact.

Fact 3.3.2. For a regular n-run design with 3 ≤ m ≤ n/2 factors, the maximum A3 is attainedonly by the designs constructed by repeating a design with m factors and n∗ runs, where n∗ = 2q

with q satisfying 2q−1 ≤ m ≤ 2q − 1. Clearly, A3 = 0 for m = 1 and 2.

By the above fact, we have the following five results:

i Design D̄ contains any one or two columns from H, if m̄ = 1 or 2;

ii Design D̄ is found from 4-run designs, if m̄ = 3;

iii Design D̄ is found from 8-run designs, if 4 ≤ m̄ ≤ 7;

iv Design D̄ is found from 16-run designs, if 8 ≤ m̄ ≤ 15;

v Design D̄ is found from 32-run designs, if 16 ≤ m̄ ≤ 31.

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According to the above results and Rule 3.3.1, we obtain the corresponding unique design D̄

of m̄ factors for 8 ≤ m̄ ≤ 31 from the complete catalogue for 16-run designs and 32-run designs.The 4-run design with 3 factors and 8-run designs with 5 ≤ m̄ ≤ 7 factors are unique for eachm̄. For m̄ = 4, there are two nonisomorphic designs, and we choose the one with large A3(D̄).The final results are summarized in Table 3.4. Tang and Wu (1996) also gave some recursiveformulas, which can be used to determine the constants in (3.1)-(3.3). After some calculations,we obtain

A3(D) = 13

[(m̄

2

)+(

m

2

)− mm̄

2

]−A3(D̄), (3.4)

A4(D) = 14

[m− 3

3

(m̄

2

)− m̄ + 1

3

(m

2

)+ mm̄

3−(

m̄

3

)+(

m

3

)]+ A3(D̄) + A4(D̄), (3.5)

and

A5(D) = 15

[(m̄

4

)+ 1

4

(m̄

3

)(4−m) + 1

6

(m̄

2

)(3m̄− 7m + 9) + 1

6

(m

2

)(m̄

2

)+(

m

4

)

−14

(m

3

)(m̄− 1) + 1

12

(m

2

)(3m̄− 4m + 13) + 1

24mm̄ (7m− m̄− 21)

]−(2f−1 −m)A3(D̄)−A4(D̄)−A5(D̄). (3.6)

Using (3.4)-(3.6), the A3, A4, A5 values of minimum aberration designs with 33 ≤ m ≤ 63 arereadily determined, and also contained in Table 3.4.

Table 3.5 provides (A3, A4, A5)’s of minimum aberration designs and those of best designsfrom our approach. The results are very satisfying. About two-thirds of our best designs givethe same (A3, A4, A5) as the corresponding minimum aberration designs. Even though somecases give different results, the discrepancies are rather small. Among those cases with differentresults, there are 9 cases giving the identical A3 values. Only 2 cases have different A3 values,but the differences are quite small.

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Table 3.4: (A3, A4, A5) of design D̄ and that of the corresponding MA design D

D̄ DNumber of Factors (m̄) (A3, A4, A5) Number of Factors (m) (A3, A4, A5)

30 (140, 945, 4368) 33 (16, 1240, 1120)29 (126, 819, 3640) 34 (32, 1256, 2240)28 (113, 706, 3012) 35 (48, 1288, 3376)27 (101, 605, 2473) 36 (64, 1336, 4544)26 (90, 515, 2013) 37 (80, 1400, 5760)25 (80, 435, 1623) 38 (96, 1480, 7040)24 (71, 365, 1292) 39 (112, 1577, 8402)23 (63, 304, 1015) 40 (128, 1691, 9860)22 (56, 251, 784) 41 (144, 1822, 11432)21 (50, 205, 592) 42 (160, 1970, 13136)20 (45, 175, 453) 43 (176, 2145, 14960)19 (41, 147, 337) 44 (192, 2334, 16960)18 (38, 126, 252) 45 (208, 2543, 19136)17 (36, 112, 196) 46 (224, 2773, 21504)16 (35, 105, 168) 47 (240, 3025, 24080)15 (35, 105, 168) 48 (256, 3300, 26880)14 (28, 77, 112) 49 (280, 3556, 29904)13 (22, 55, 72) 50 (304, 3836, 33184)12 (17, 38, 44) 51 (328, 4140, 36744)11 (13, 25, 25) 52 (352, 4468, 40608)10 (10, 15, 12) 53 (376, 4820, 44801)9 (8, 10, 4) 54 (400, 5199, 49344)8 (7, 7, 0) 55 (424, 5603, 54264)7 (7, 7, 0) 56 (448, 6034, 59584)6 (4, 3, 0) 57 (476, 6482, 65240)5 (2, 1, 0) 58 (504, 6958, 71344)4 (1, 0, 0) 59 (532, 7462, 77924)3 (1, 0, 0) 60 (560, 7995, 85008)2 (0, 0, 0) 61 (590, 8555, 92568)1 (0, 0, 0) 62 (620, 9145, 100688)0 (0, 0, 0) 63 (651, 9765, 109368)

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Table 3.5: (A3, A4, A5)’s of MA designs and those of the corresponding best designs from ourapproach for 33 ≤ m ≤ 63

Number of Factors MA designs Constructed designs Number of Factors MA designs Constructed designs(m) (A3, A4, A5) (A3, A4, A5) (m) (A3, A4, A5) (A3, A4, A5)33 (16, 1240, 1120) (16, 1240, 1120) 49 (280, 3556, 29904) (294, 3479, 29841)34 (32, 1256, 2240) (32, 1256, 2240) 50 (304, 3836, 33184) (304, 3836, 33184)35 (48, 1288, 3376) (48, 1288, 3376) 51 (328, 4140, 36744) (328, 4140, 36744)36 (64, 1336, 4544) (64, 1337, 4544) 52 (352, 4468, 40608) (352, 4469, 40608)37 (80, 1400, 5760) (80, 1401, 5760) 53 (376, 4820, 44801) (376, 4821, 44800)38 (96, 1480, 7040) (96, 1483, 7040) 54 (400, 5199, 49344) (400, 5199, 49344)39 (112, 1577, 8402) (112, 1579, 8400) 55 (424, 5603, 54264) (424, 5603, 54264)40 (128, 1691, 9860) (128, 1694, 9856) 56 (448, 6034, 59584) (448, 6034, 59584)41 (144, 1822, 11432) (144, 1822, 11432) 57 (476, 6482, 65240) (476, 6482, 65240)42 (160, 1970, 13136) (160, 1970, 13136) 58 (504, 6958, 71344) (504, 6958, 71344)43 (176, 2145, 14960) (176, 2146, 14960) 59 (532, 7462, 77924) (532, 7462, 77924)44 (192, 2334, 16960) (192, 2335, 16960) 60 (560, 7995, 85008) (560, 7995, 85008)45 (208, 2543, 19136) (210, 2520, 19215) 61 (590, 8555, 92568) (590, 8555, 92568)46 (224, 2773, 21504) (224, 2773, 21504) 62 (620, 9145, 100688) (620, 9145, 100688)47 (240, 3025, 24080) (240, 3025, 24080) 63 (651, 9765, 109368) (651, 9765, 109368)48 (256, 3300, 26880) (256, 3300, 26880)

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Chapter 4

Concluding Remarks

Starting with any two fractional factorials, either regular or nonregular, D1 with m1 factorsand n1 runs and D2 with m2 factors and n2 runs, we present a general approach to constructinga large design D with m = m1m2 factors and n = n1n2 runs. For any k with 1 ≤ k ≤ m1m2,based on the type of D1 we derive three equations that connect Bk(D) to Bs(D1)’s and Bt(D2)’sfor s, t ≤ k. These results imply that the best design D from our approach can be constructed bychoosing two minimum G2-aberration designs D1 and D2. Regular 64-run designs of 7 ≤ m ≤ 63factors are used to evaluate this approach. The findings are very promising - the A3, A4, A5 valuesof the best designs from our approach are the same as or very close to those of the correspondingminimum aberration designs.

In this project, the evaluation of our approach focuses on designs of 64 runs. One futurework would be to evaluate our approach by looking at designs with 128 runs, as all minimumaberration designs of 128 runs are known, which can be used to compare with designs constructedfrom our approach. On the other hand, Deng and Tang (1999) defined another generalizationof minimum aberration criterion, minimum G-aberration, to compare factorials for a givennumber of factors and run size. We may then use the general construction to obtain minimumG-aberration designs, which would be another interesting topic for the future work.

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[3] Chen, J., Sun, D. X., and Wu, C. F. J. A catalogue of two-level and three-levelfractional factorial designs with small runs. Int. Statist. Rev. 61 (1993), 131–145.

[4] Deng, L. Y., and Tang, B. Generalized resolution and minimum aberration criteria forPlackett-Burman and other nonregular factorial designs. Statist. Sinica 9 (1999), 1071–1082.

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[6] Tang, B. Orthogonal arrays robust to nonnegligible two-factor interactions. Biometrika93 (2006), 137–146.

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